where \(a\)  has been defined in (REF ). The polynomial \(p_{k}(x)\)  of degree \(2k\)  is related to different representations of the standard Jacobi polynomial [1]}, [2]}
\(J^{(a,b)}_{k}(x)&=&\frac{(-1)^{k}(b+1)_{k}}{k!}\sum _{i=0}^{k}\frac{(-k)_{i}(k+a+b+1)_{i}}{(b+1)_{i}\Gamma (i+1)}\left(\frac{1+x}{2}\right)^{i}\\&=&\frac{1}{k!}\sum _{i=0}^{k}\frac{(-k)_{i}(k+a+b+1)_{i}(i+a+1)_{k-i}}{\Gamma (i+1)}\left(\frac{1-x}{2}\right)^{i}\\ &=&\sum _{i=0}^k \frac{(-1)^i \Gamma (a+k+1) (k+b-i+1)_i}{\Gamma (i+1) \Gamma (a+i+1) \Gamma (k-i+1)}\left(\frac{1-x}{2}\right)^i \left(\frac{1+x}{2}\right)^{k-i}\) 
